Mathematical Tools for Physics

14—Complex Variables 444

By assumption,fis bounded,|f(z)|≤M. A basic property of complex numbers is that|u+v|≤|u|+|v|for
any complex numbersuandv. This means that in the defining sum for an integral,

∣

∣

∣

∣

∣

∑

k

f(ζk)∆zk

∣

∣

∣

∣

∣

≤

∑

k

∣

∣f(ζk)

∣

∣

∣

∣∆zk

∣

∣, so

∣

∣

∣

∣

∫

f(z)dz

∣

∣

∣

∣≤

∫

|f(z)||dz| (16)

Apply this.

|f(z 1 )−f(z 2 )|≤

∫

|dz||f(z′)|

∣

∣

∣

∣

z 1 −z 2

(z′−z 1 )(z′−z 2 )

∣

∣

∣

∣≤M|z^1 −z^2 |

∫

|dz|

∣

∣

∣

∣

1

(z′−z 1 )(z′−z 2 )

∣

∣

∣

∣

On a big enough circle of radiusR, this becomes

|f(z 1 )−f(z 2 )|≤M|z 1 −z 2 | 2 πR

1

R^2

−→ 0 asR→∞

The left side doesn’t depend onR, sof(z 1 ) =f(z 2 ).